Modeling diffusive transport with a fractional derivative without singular kernel

In this paper we present an alternative representation of the diffusion equation and the diffusion-advection equation using the fractional calculus approach, the spatial-time derivatives are approximated using the fractional definition recently introduced by Caputo and Fabrizio in the range beta, gamma is an element of (0; 2] for the space and time domain respectively. In this representation two auxiliary parameters sigma(chi) and sigma(t) are introduced, these parameters related to equation results in a fractal space-time geometry provide an entire new family of solutions for the diffusion processes, The numerical results showed different behaviors when compared with classical model solutions. In the range beta, gamma is an element of (0; 1), the concentration exhibits the non-Markovian Levy flights and the subdiffusion phenomena; when beta = gamma = 1 the classical case is recovered; when beta, gamma is an element of (1; 2] the concentration exhibits the Markovian Levy flights and the superdiffusion phenomena; finally when beta = gamma = 2 the concentration is anomalous dispersive and we found ballistic diffusion. (C) 2015 Elsevier B.V. All rights reserved.